Taken originally from Scala, Manuel Op De Coul, this is a user-editable list of modes of 31edo. The numbers indicate successive intervals, in dieses. Strictly proper scales of seven notes can be found here.

Nothing special

Or, to some, the only important thing.

2 3 3 2 3 2 3 2 3 3 2 3	Meantone Chromatic (53/220-comma)
5 5 3 5 5 5 3	Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian
5 3 5 5 3 5 5	Thirty-one tone Natural Minor, Intense Diatonic Hypodorian, Aeolian
5 3 5 5 5 5 3	Thirty-one tone Melodic Minor
5 3 5 5 3 7 3	Thirty-one tone Harmonic Minor
5 5 3 5 3 7 3	Thirty-one tone Harmonic Major
5 5 3 5 3 5 5	Thirty-one tone Major-Minor
5 2 6 5 2 5 6	"Septimal" Natural Minor

Euler-Fokker genera

Boxy shapes from the [3,5,7] lattice.

5 8 5 13	Genus primum
10 3 5 5 5 3	Genus secundum
8 2 8 3 7 3	Genus tertium
10 10 10 1	Genus quartum
5 7 6 7 5 1	Genus quintum
4 6 2 6 4 3 3 3	Genus sextum
4 6 5 6 4 6	Genus septimum
6 6 6 1 6 6	Genus octavum
4 6 9 6 4 2	Genus nonum
13 6 6 6	Genus decimum
5 5 3 5 5 3 2 3	Genus diatonicum
3 5 2 3 5 3 2 5 3	Genus chromaticum
5 5 2 1 5 5 2 3 3	Genus diatonicum cum septimis
3 4 3 3 2 1 4 1 4 1 2 3	Genus enharmonicum vocale
2 2 4 2 2 3 3 3 1 3 3 3	Genus enharmonicum instrumentale
3 2 3 2 3 2 3 3 2 3 2 3	Genus diatonico-chromaticum
5 2 1 2 5 3 2 1 4 1 2 3	Genus bichromaticum
3 1 2 4 2 4 2 4 2 1 3 3	Genus [3577] (Joel Mandelbaum Andante cantabile)

Greek/tetrachord modes

4 4 5 4 4 5 5	Neutral Diatonic Mixolydian
4 5 4 4 5 5 4	Neutral Diatonic Lydian
5 4 4 5 5 4 4	Neutral Diatonic Phrygian
4 4 5 5 4 4 5	Neutral Diatonic Dorian
4 5 5 4 4 5 4	Neutral Diatonic Hypolydian
5 5 4 4 5 4 4	Neutral Diatonic Hypophrygian
5 4 4 5 4 4 5	Neutral Diatonic Hypodorian
4 5 4 4 5 4 5	Neutral Mixolydian
5 4 4 5 4 5 4	Neutral Lydian
4 4 5 4 5 4 5	Neutral Phrygian
4 5 4 5 4 5 4	Neutral Dorian
5 4 5 4 5 4 4	Neutral Hypolydian
4 5 4 5 4 4 5	Neutral Hypophrygian
5 4 5 4 4 5 4	Neutral Hypodorian
2 2 9 2 2 9 5	Hemiolic Chromatic Mixolydian
2 9 2 2 9 5 2	Hemiolic Chromatic Lydian
9 2 2 9 5 2 2	Hemiolic Chromatic Phrygian
2 2 9 5 2 2 9	Hemiolic Chromatic Dorian
2 9 5 2 2 9 2	Hemiolic Chromatic Hypolydian
9 5 2 2 9 2 2	Hemiolic Chromatic Hypophrygian
5 2 2 9 2 2 9	Hemiolic Chromatic Hypodorian
2 3 8 2 3 8 5	Ratio 2:3 Chromatic Mixolydian
3 8 2 3 8 5 2	Ratio 2:3 Chromatic Lydian
8 2 3 8 5 2 3	Ratio 2:3 Chromatic Phrygian
2 3 8 5 2 3 8	Ratio 2:3 Chromatic Dorian
3 8 5 2 3 8 2	Ratio 2:3 Chromatic Hypolydian
8 5 2 3 8 2 3	Ratio 2:3 Chromatic Hypophrygian
5 2 3 8 2 3 8	Ratio 2:3 Chromatic Hypodorian
3 5 5 3 5 5 5	Intense Diatonic Mixolydian, M.Locrian
5 3 5 5 5 3 5	Intense Diatonic Phrygian, M.Dorian
3 5 5 5 3 5 5	Intense Diatonic Dorian, M.Phrygian
5 5 5 3 5 5 3	Intense Diatonic Hypolydian, M.Lydian
5 5 3 5 5 3 5	Intense Diatonic Hypophrygian, M.Mixolydian
2 5 6 2 5 6 5	Soft Diatonic Mixolydian
5 6 2 5 6 5 2	Soft Diatonic Lydian
6 2 5 6 5 2 5	Soft Diatonic Phrygian
2 5 6 5 2 5 6	Soft Diatonic Dorian
5 6 5 2 5 6 2	Soft Diatonic Hypolydian
6 5 2 5 6 2 5	Soft Diatonic Hypophrygian
5 2 5 6 2 5 6	Soft Diatonic Hypodorian
1 2 10 1 2 10 5	Enharmonic Mixolydian
2 10 1 2 10 5 1	Enharmonic Lydian
10 1 2 10 5 1 2	Enharmonic Phrygian
1 2 10 5 1 2 10	Enharmonic Dorian
2 10 5 1 2 10 1	Enharmonic Hypolydian
10 5 1 2 10 1 2	Enharmonic Hypophrygian
5 1 2 10 1 2 10	Enharmonic Hypodorian

Of particular interscalar interest

6 6 7 6 6	Quasi-equal Pentatonic
3 3 2 3 3 3 3 3 2 3 3	near 11edo

Miscellaneous

3 2 2 3 3 2 3 3 2 2 3 3	Fokker 12-tone
5 3 5 3 5 2 5 3	Modus conjunctus
3 5 2 5 3 5 3 5	Octatonic
3 3 4 3 5 3 4 3 3	Hahn symmetric pentachordal
3 4 3 3 5 3 4 3 3	Hahn pentachordal
4 4 2 5 3 3 4 3 3	Hahn Nonatonic
5 3 3 3 3 5 3 3 3	Rothenberg Generalized Diatonic
(5) 3 1 4 (5) (5) 3 1 4	Michael Sheiman's "Dimension" scale (Maqam Rast mixed with Meantone Diatonic...lattice (ratio/chord map) available below
(3 2)3 1 4 (3 2)(3 2) 3 1 4	Michael Sheiman's extended 12 tone "Dimension ECT" scale (Maqam Rast mixed with Meantone Diatonic and Michael Sheiman's "6-factor" no wolves 12th harmonic scale)

MOS and Cradle MOS

4 5 4 5 4 5 4	Neutral scale 7-note
4 1 4 4 1 4 4 4 1 4	Neutral scale 10-note
4 1 4 4 4 1 4 4 1 4	Breed 10-tone
4 3 4 3 4 3 4 3 3	Orwell[9]
5 1 5 1 5 1 5 1 5 1 1	de Vries 11-tone
3 1 3 1 3 1 3 1 3 1 3 1 3 1 3	Pseudo-Porcupine 15-note
1 7 1 2 1 7 1 2 1 7 1	Nested Bees
4 2 4 2 4 2 4 3 3 3	Lumma Decatonic
2 5 2 2 5 2 2 2 5 2 2	Secor/Barton no-fives